Discharge Calculation in Interfering Walls By Modified Total Drawdown-Discharge Eguations

Discharge Calculation in Interfering Walls By Modified

Total Drawdown- Discharge Eguations

Geliştirilmiş Toplam Düşüm-Debi Denklemleri Yardımı ile

Girişim Yapan Kuyuların Debilerinin Hesaplanması

Ahmet Hikmet ÖZYOL 1

1 Water Supply Department of the Holy City of Makkah.

Bu araştırmada, girişim yapan kuyularda, farklı düşim, rurKiı ya­ rıçap ve farkiı çalışma süresi olması halinde kuyuların debilerinin he­ sabı için geliştirilmiş toplam düşüm - debi denklemlerinden faydalanıl- mıştır.

Ayrıca basmçsız akiferdeki kuyular için tam toplam düşüm debi denklemi kullanılmıştır.

In this research it was attempted to calculate discharges of inter- fering wells in the case of different drawdowns, diameters and opera- ting periods by means of modified total drawdown discharge equations.

Also exact total drawdcwn -discharge equation was used for the wells which are in unconfined aquifers.

INTRODUCTION

Muskat (1) used total drawdown - discharge eauations to calcula­ te discharges of interfering wells for steady flows in the case of equal drawns, diameters and operating periods, and furthermore gave some special Solutions. Hovvever, Hantush (2) used similar eauations for un- steady flows and presented some special equations with the same con- ditions.

128 Ahmet Hikmet özyol

DEKIVATION OF EQUATIONS

Muskat cmployed the following equations for diseharge calcula- tions in the interfering vvells for the case of steady flovvs as,

n

iT=H-hT=y -7-■ HfKi'ni / , 2, îtk. ₺

i- ı n

H2 — h2= ■ İniRî/rj ı«-l

(1)

(2)

Eqs. (1) and (2) are valid for confined and unconfined aquifers respec- tively. Eq. (1) was originally derived on the basis of superposition prin- ciple as,

n

8t= V S; (3)

i—1

but Eq. (2) is an appro;ximate equation and is valid only for small va- lues of drawdoıvns (sT«2/f), where

sT : Total drawdown in a iveli, L.

hT : Height of wateı- in a well which corresponds to total draw- doA'n in the same well, hT=H—sT, L.

Sj : Individual drawdown in the i—th well, L.

H : Height of piezometric pressure froın the base of a confined aquifer or thickness of an unconfined aquifer L.

b : Thickness of a confined aquifer, L.

k : Coefficient of permeability, L T.

Q, : Diseharge of the i—th well, LP/T.

r{ : Radius of the i—th well, L.

R, ; Radius of influence in the i— th well, L.

In the solution of Eqs. (1) and (2) Muskat accepted that draw- dovvns, diaıneters and operating periods are the same in ali the inter­

fering tvells. In this research, it vvas attempted to find Solutions for different drawdowns, diameters as well as operating periods for each individual interfering wells. Also, exact total drasvdown eauation was

Dlscharge Calculation in Interfering VVells By Modified Total... 129

used, Eq. (3), for wells in unconfined aguifers instead of approximate equation, Eq. (2). First of ali Eq. (3) is modified and written in a new form as,

n

STİ=2\- (i = l, 2, ...n) (4)

İ=1 where

sTİ : Total drawdown in the i—th well, L.

Sjj : Influence (drawdown) in the i—th well which is caused by İ—th well, L.

Eq. (4) can be written explicitly as,

Sil = Su + Sn + ... + 8 in

STî = ®21 "t + ••• + Sjn

I

(5)

«Tn = «ol + + ••• +«»□

However, for confined aquifers (in the case of steady flow) mutual drawdown effects can be eacpressed as,

İn (Rî/rij) M

(6)

and for unconfined aquifers this eotpression turns out to be,

Sii = H— *

/ r ’ - . q

V 75 . fc (7)

By defining a new variable as,

İniRj/rtj) aı,“ 2.K.k.b and

İn (Rı/rii)

3"= *JT~

Eqs. (6) and (7) can be written implicitly as,

(8)

(9)

ISO Ahmet Hikmet özyol

»ii =a<î • Q; (10)

and

sti= H — \/H2 — . Qj (İD

where

Tu : The distance between the i*— th and j—th well, L.

Rt : Radius of influence in the j—th well, L.

Çj : Discharge of the j—th well, LP/T.

p,J | : Dummy variables.

If Eq. (10) is substituted into Eq. (5), then for confined aquifers one can find,

sti = <xh . Q] +otiî. Q} + ... +«10.Qn j

STJ a21 • Ql + a22 • Q?+ ••• + a2n • Q" İ. Q2) STo = <Xnl • Q1+ ®n2 • Q2 + • • • + «nn • Qn Î

or shortly,

n

STi=^aiı Qj H = l, 2,... ,n) (12a) j=l

However, for unconfined aquifers, first, it is useful to define

Sli = y/ H - Pij . Q; (13) and accordingly Eq. (11) becomes,

S|;=H—S;j (14)

On the other hand, if Eq. (14) is substituted into Eq. (5) it leads to,

sti—n. H — (8n+ 613 + ... + 810) ST2 = n.H—(821+Sn+ ... +82n)

std = m • H — (8„ı + Sn2+ ... + 80n) j

(15)

or

Discharge Calculation in Interfering Wells By Modified Total... İSİ

or n.H— ^«ti = 5ıı+812+ ••• + 8In n.H—«T:= 831 +S22+ ... 4-52n n.H—Sfn = 80I +8n2 + ... +Son

(16)

Furthermore, it can be revvritten shortly as, n

n.H-ST^V&i (t = 1,2...n) (16 a) j=l

Since, 5ij is an irrational function, it is necessary to use Computer for numerical Solutions of Eq. (16).

However, modified approximate eguations can be used also for con- fined aquifers in the case of different drawdowns, diameters, and ope- ration periods provided that drawdowns are small.

Modified approximate general eauation can be written as follows,

H2—h’Tİ=V ln-(Ri,rip • Qj a fC

j-1

under the light of Eq. (9), Eq. (17) can be rewritten as, n

H’-h’n^Pu.Ç;

or explicitly as, j=l

(i=l, 2...n) (18)

H2-VT1= 3ıı•Qı +3)2 Q2+...+3,n.Qn H2— h2T2 =021 • Ql + p22 • Q? + 02“ ■ Qo

(17)

(19)

H2 - fo2T. = 3m.Q,+3,2 .Qî+ ... + 3nn. Qo

Also, Eq. (19) takes the follovving form of equation system, M,— 3„ . Q, +3n ■ Q2 + ••• 3ı» • Q» 1

m2=32i . Qı + 322■ Qz — +0z" Q» i

...I

Mn = 3nl.Q1 + 3.2.Qî+...+ 3on.Qn l which can be written briefly as,

(20)

132 Ahmet Hikmet özyol

Mi = 3i| ♦ Q|

j=ı

(i = 1,2...n)

vvhere in ali the above equations

Mi=H2 —h’TI

Eq. (20) is similar to Eq. (12) and can be solved easily.

On the other hand, for unsteady flow in confined aquifers be written from Theis (3) equation as,

W (w, j) _ S|‘ 4.7t.fc.b ' '

with the definition of the follovving new variables, W(«||)

4. n . k.b

Eq. (22) can be conciesly written as,

»îi=aii • Qi

And for unsteady flow in confined aquifers from modified Theis tion one can write,

H

2 =

? .

ft can be written that,

s =H—ı /h

2 —

V 2. k .k Q due to h=H—s, or

W. ^q .

2. it. k

(21)

(21a)

it can

(22)

(23)

(24)

equa-

(25)

(26) with the definition of the following new variable

• o w(U|j)

Pii“ 2.ıt.fc Eq. (25) can be written briefly as,

Discharge Calculatton in Interferlng Wells By Modified Total... 133

31İ = H-V'h2-3.İ.Qİ (27)

And also if it is defined that,

5,1= VH’-pn.Çi (28)

Eq. (27) becomes as,

<29) In this case Eqs. (12), (16) and (20) can be used for unsteady flows by replacing a,j, fîıj, 5ıj instead of <Xîj,0ij, Sı,. The meanings of some parameters in the above derivations are as follows,

W(Wij) : Well function.

«ij : S-r3ij/4-T.fo •

r,j : Distance between the i— th and j—th wells, L.

S : Storage coefficient.

T : Transmissibility. Lr/T.

a,<2 ' ] : Dummy variables.

8,j ’

: operation time, T.

APPLICATIONS Problem I :

4 wells in a confined aquifer which field in the case of steady flow.

are randomly scattered in the

4

134 Ahmet Hikmet Özyol

Solution :

First, Eq. (12) is written

Sti=011 • Ql+'a12 • Q2 +a13 ■ Qs4" a14 • Ql

STî= «21 ■ Q1 4"«22 • Qî 4" «23 • Qs4"a24 • Q4 •

Stj=OC31 ■ Qı 4-0132 ■ Q24-<*33 • Qs4- a34.Q4 ST4 =«41 ■ Ql4- OC42. Q?4"«43 • Ql 4" a44• Q4

(i-D

Then according to Eq. (8) the values of a,j are calculated as follows, an= ln (R^rnl/2.n.k.b

a12 = İn (R2/tu)/2 . ti. k.b

«43=ln (R3/r13)/2.n.k.b

«44 =İn (Rt/rtt)/2 .ıt.k.b

The values sTı , «T2, sT3 ,sT4 are given in the beginning as data. For a special case, if it is assumed that four wells are on the comers of a square, and drawdowns, diameters and operation periods are equal to cach other in the well group as shovvn in the follovving sketch, the cal- culations can be achieved as follows

Due to the symmetric well distribution one can write the follovving points :

Dlscharge Calculatlon in Interfering Welis By Modified Total... 135

1) r]2_r23— — r41 , rıs — 2) rn= rn - r33 = r44= t

3) Sti =3t2 = Sts = 8t4 = 3t

4) tı = t2 =t3=f4= t or

Rj — R2 — R3 “ R4 = R

From the above mentioned knowledge the follovving results are found, 1 2

«ll =«22 =«S3= a44 = 2 Tt k.b and

a12=«23=«34=a4I) = —1 ın(R/L)

Otjj— — ®43 —^14 ) 2 • K■ rC .u and

«İS =«31 =«24 = «42= • İD <R//2 • L)

After these procedures ali a>j values are substituted in Eqs. (1-1) to- gether vvith STi=«T2=3T3=3i4=Sr. and because of the symmetry, it can be written Q! — 0^=03=Q4 = Q. Hence, it is found that

sT= - [İn (R/r) 4- In (R/L)+ İn(R/V 2 . L) +İn (R/L)]

2^.it.k. b

where in Eq. (I -1) each of the four equations become the same. Hence, it is found that,

In fact, this is the same eguation that was found by Muskat (1) in a confined aquifer, in the of case four wells which are on the comers of a square, for steady fknv.

Problem II :

3 wells in a confined aquifer which are randomly scattered in the field in the case of steady flow

136 Ahmet Hikmet özyol

Solution :

First, Eq. (16) is written

3 .H — 3Ti = 8n+ 6n +8]3 I

3 .H — Sti = Sji+5j2 + 8,5 (II—1J 3 .H—873= 831+832 + 855 I

where from Eq. (13)

6ü= VH»—(II-2)

and from Eq. (9)

ft^a/ıt.fcJ.hHRj/nj) (II-3)

From Eq. (II -3) it can be seen that :

3n=(l/K.k).ln(Rı/ru) 3,2 = (1/7C. k). ln(R2+12) 0IJ =(l/ır.k). İn(Rs/r13) 0M=(1/K.k).ln (R3/r33)

Due to the fact that 6ıj is irrational function of Qj, it is necessary to use Computer for numerical Solutions. If it is desired to use approKİ- mate equation for small drawdowns, Eq. (19) can then be written as,

HJ—h‘Tı = 0n ,Q, + 0I2,Q2+3n. Qs |

H’-k’T2 = 0ji.Qi + 0m.Q2+023.Q3 ' (II-4) H1— h2T3=33ı • Q1+032 • Q2 +033-Qa )

The values of are the same as before.

For a special case, where three wells are on the corners of an equi lateral triangle and drawdowns, diameters and operation periods are equal then it is possible to write the following points.

Dischnrge Calculation in Interferinç Wells By Modified Total.. • L37

D rn = r73=^{r31 =}L

2) rn= r22= r3S=r

3) hTl = hT2 = ^T3=Ht

4) t,=t2=/3=t or Rı —— R2 —Rg—R

From the above mentioned knowledge the following results are found.

3ıı~ 322 — 0s3— n ^-ln(R/r) 312— 323—3si

6si3 P32= 313 1

= -iy • İn(R/L) İt.K

Af ter these calculations ali 3u valuea are substituted into Eq. (II- 4) together with

/»Tl = hTı =hT3 — ht

Also, it is a fact that Qı—Q2=Qr\~Q which are due to symmetry. The­

se considerations leads us to,

H2— h2T= ~ • [ln(R/r) + ln (R/L) + İn (R/L)]

K • K

where, in Eq. (H-4) system, each of the three eauations become the same. Hence, it is found,

tc . k. (H2 — h2r) (H-5)

W in(Rs/L2,r) ' ’

This is the same equation that was earlier found by Muskat in an un- confined aquifer, for three wells which are located on the comers of an eguilateral triangle, in the case of steady flow.

Havvever, for this special case exact equations, Eqs. (II-1) and (II - 2) can be solved without going to Computer. First of ali it is bet- ter to write 8ıj values

811= VH’-fln.ç, 8„=

833 = \/ H2 3s3 • Qs For this special case one can write,

138 Ahmet Hikmet özyol

and also because Qı = Q2=Q3=Ç. due to symmetry,

0iı — 022 — 0»= ~ ■ İn (R/r) —

And also it is given that, sTi=St2='St3=St, hence it can be written that, 511=5„ = 5n= \/H2 —£r.Q =S,

Furthermore,

312 = 023 =031

/=1.1

(R/L) = P

021 =032=013

and therefore

8„-523-5SI/ = v

/

2_P

.

=5

5n - 832 — 813'

or explicitly

3.H -ST = 6r+ 2.5L (II-6)

3 .H —sT = \/H2— |3r .Q+2 . H2 — .Q (H-7)

where, in Eq. (II-1) system, each of the three equations become the same. After solution Eq. (II- 7) it is found that,

vvhere

q

=(—

,±\/

’3-4

3.

3)'2.

3 ui-a>

A3 = fJr ■ 0L — E’j

B3 =—(H2.0r + H2.

0

4-2. E

and

E

= [(3.H-

)2-5.H2]/4

E

=

The problem of two interfering wells with equal drawdowns can be sol- ved in the same way,

where

2' ^h T ‘ ^s "^'^ VI9'

4 « İl ST2 ” ®21 ’ ^22 \

Discharge Calculation in Interfering Wells By Modified Total ... 139

5ıı= 822 =8r S12 =S2] =8l

8ti=ST2 = ST Hence, it is found

2.H — ST = 8r + 8L (11-10) since

6,= v^H2—0, .Q 5l= 3l.Q it can be vvritten

2.H—«T= .Q + v/H2— |Îl.Q After solution Eq. (II -11) system it is found that

Q =(—B± y^B22 —4.A2.C2 )/2.A2 vvhere

A2=3,.3l-F’2

B2 = — (H2.0r + H2.Pl + 2. F,. F2)

c2=h« — f,2 and

Fj =[(2.H— sT)2 — 2 . H2]/2 F2=(0r +3J/2

R E F E R E N C E S

1. Muskat, M., The Flow of Homogeneous Fluids through Porous Medla, McOraw - Hlll Book Company, New York, 1937.

2. Hantush, M. S., Hydraulics of Wells, İn V.T. Chow (ed.), Advances in Hydro- science, Vol. 1, Academic Press, New York, pp. 281 - 432, 1964.

3. Thels, C. V., The Relatlon betwcen the lowerlng of the Piezometric surface and the rate and duratlon of discharge of a well using groundwater storage, Trans.

Amer. Geophys. Union, Vol. 2, pp. 519 - ü24, 1935.

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